High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

Abstract EN

The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy.

In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation.

Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme.

Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique.

Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.